Improving Pedagogical Content Knowledge On Rational ...

196 Global Education Review 5(3)

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link to the license is provided, and you indicate if changes were made. Citation: Van, Leap, Mao, Sokalyan, & Cnudde, Veerle (2018). Improving pedagogical

knowledge on rational numbers of Cambodian teacher trainers. Global Education Review, 5 (3), 196-211.

Improving Pedagogical Content Knowledge

On Rational Numbers

Of Cambodian Teacher Trainers

Leap Van

VVOB Cambodia

Sokalyan Mao

VVOB Cambodia

Veerle Cnudde

VVOB Cambodia

Abstract

Despite adequate facilities and several education reforms, most Cambodian teacher trainers fail to provide

sufficient content knowledge and student-centered pedagogy. Many also lack the skills to diagnose pre-

service teachers’ misconceptions and to propose adequate solutions. Dictating lessons with little feedback

or applied activities or having pre-service teachers copy off the board for extended periods, suggests low-

quality instruction (Tandon & Fukao, 2015). To tackle this, the Flemish Association for Development

Cooperation and Technical Assistance (VVOB- education for development)1 developed a 3-year (2014-

2016) programme in close collaboration with the Cambodian Ministry of Education, Youth and Sport

(MoEYS). The programme was rolled out in all primary teacher training colleges (PTTCs). One of the

interventions in this programme aimed at improving both Pedagogical Content Knowledge (PCK) and

Content Knowledge (CK) on rational numbers of mathematics teacher trainers, with a focus on 1)

mathematics content knowledge, 2) the use of representations to enhance pre-service teachers’

understanding, 3) assessing pre-service teachers’ learning, and 4) addressing misconceptions. A total of

54 mathematics teacher trainers participated in this intervention. Their capacity was built through

training, coaching, mentoring and try-outs with pre-service teachers. The impact of the intervention was

measured through a pre-test post-test design, enriched by qualitative data collected during 97 lesson

observations. After the intervention, 91% of the teacher trainers had significantly increased their score on

the PCK test and 94 % had improved their teaching strategy in at least two of the three criteria of PCK. In

this paper, the design and impact of the intervention are explained, and suggestions for further research

are provided.

Keywords

Content Knowledge, Pedagogical Content Knowledge, Teacher Education, Misconceptions, Coaching,

Mentoring, Teacher Trainer, Mathematics, Pre-service teachers

Introduction

Since the end of the Pol Pot regime, the

curriculum of general education in Cambodia

has gone through several major reforms. In the

early 1980s, Cambodia’s education systems were

restructured, and this progress was marked as

the country’s recommitment to socio-economic

development and expanding educational

opportunity (Dy, 2004). ________________________________________

Corresponding Author

Leap Van, VVOB Cambodia, Phnom Penh Centre Corner of

Sihanouk Blvd and Sothearos Street Building F, Room 273

(2nd floor) Phnom Penh, Cambodia

Email: [email protected]

mailto:[email protected]

Pedagocical content knowledge of Cambodian teacher trainers 197

In the period 1979-1986, the general

education system consisted of 10 years (4+3+3):

four years of primary education, three years of

lower secondary education, and three years of

upper secondary education (Hang, 2016). In the

1986-1996 period, the system changed into an

11-year (5+3+3) trajectory. From 1996 onwards,

the general education system contained 12-years

in school (6+3+3). There has been a shortage of

qualified teachers throughout these reforms and

the recruitment of teachers and teacher trainers

was not well-structured.

As stated in Hang (2016), teacher training

in the early eighties consisted mainly of short

courses to upgrade the knowledge of former

teachers, students and educated individuals who

had survived the Pol Pot regime. The duration of

these training courses varied between one and

three months. In 1983, the Ministry of

Education published teacher standards for

preservice preschool and primary schools. In the

first phase, becoming a primary school teacher

involved the completion of grade 7 followed by

one year of teacher training. Due to the lack of

teachers, these standards were reduced in some

disadvantaged and remote areas to one year of

teacher training after completing grade 5 or even

grade 3.

Between 1986 and 1996 the requirements

for graduating as a primary school teacher were

changed: pre-service teachers needed at least

nine years of basic education to enter a two-year

teacher training course, for lower secondary it

was 11 years plus 2 additional years, and for

upper secondary teachers a foundation of 11

years schooling was needed to enter a 3 -year

teacher training course.

Since 1996, the trajectory to become a

primary school teacher is 12 years of general

education (9 years for disadvantaged and remote

provinces) followed by 2 years of teacher

training. Teacher Training Centers (TTCs) in

Cambodia are comprised of four categories: (1)

teacher training for pre-school teachers at the

Pre-School Teacher Training Center (PSTTC);

(2) teacher training for primary school teachers

at Provincial Teacher Training Centers (PTTCs);

(3) teacher training for lower-secondary teachers

at Regional Teacher Training Center (RTTC);

and (4) teacher training for upper secondary

teachers at the National Institute of Education

(NIE).

The Teacher Policy Action Plan (MoEYS,

2015) is a multiyear plan intended to bring

Cambodian education into the 21st century. This

ambitious plan includes, among other changes, a

reform of the teacher training curriculum into a

4-year bachelor, the development of Teacher

Educator Provider Standards and the

establishment of a Teacher Career Pathway, all

elements in the educational reform intended to

bring Cambodian education closer to the

inspiring level of several Association of

Southeast Asian Nations (ASEAN)2 countries.

Due to the impact of Pol Pot’s regime and

the above described reforms in general

education and teacher training, the background

of today's teacher trainers at the PTTCs is very

diverse. Some of the teacher trainers started

their career as primary school teachers before

entering PTTC. Others finished only lower

secondary school (grade 7, or 8 or 9) while some

finished upper secondary school (grade 11 or 12)

and graduated from the two-year teacher

training programme from either PTTCs or

RTTCs. Some teacher trainers graduated from

university with a bachelor’s degree and

continued a one-year pedagogical training at

NIE. These different levels of qualifications are

also reflected in teacher trainers’ understanding

of math. Research shows different levels of CK

and PCK. Literature suggests that to provide

insightful instruction, CK is not sufficient; it

requires PCK, which involves teachers’

understanding which combines knowledge of

subject content, of students’ understanding, and

of pedagogy (Ball, Lubienski, & Mewborn, 2001;

Baumert et al., 2010; Kunter et al., 2013; Rowan

et al., 2001).

In addition, there is a significant


relationship between the PCK of primary school

teachers’ and grade 3 learners’ achievements in

Cambodia (Ngo, 2013). PCK of teachers has the

largest impact (of several elements defined as

part of ‘teacher quality’) on learning outcomes,

even when you control for learner and school

characteristics. However, in the lowest quintiles

of pupil scores, teacher quality is not as

significant as student background or school

characteristics in predicting student

achievement. These findings strongly suggest

that, compared to other elements of teacher

quality, teacher PCK is a strong predictor of

learners’ achievement in mathematics. Learners

will benefit from having a teacher who is able to

identify pupil errors and who has deeper

knowledge of mathematical reasoning. In

addition, previous studies suggest that teacher

training and professional development system

for teacher trainers strengthen both subject and

PCK (Benveniste, Marshall, & Araujo, 2008;

Kleickmann et al., 2013).

Despite the evidence of the importance of

teachers’ PCK for pupils’ learning outcomes, the

PCK of mathematics teacher trainers in

Cambodia was reported to be very limited

(Tandon & Fukao, 2015). Tandon and Fukao

(2015) also found that many teacher trainers had

even lower knowledge of math than grade 9

pupils, which resulted in limited capacity to

diagnose students’ mistakes and to generate

effective learning of future teachers. An

essential teacher ability is to understand

students’ mathematical thinking, including

common errors made by students, and the

importance of students’ misconception of their

progress and achievement in the test (Hill, Ball

and Schilling 2008; Sarwadi & Shahrill, 2014).

Many teacher trainers did not have the skills to

diagnose misconceptions and to propose

adequate solutions for their pre-service teachers.

Low quality instructional methods are still used

by many teacher trainers, such as dictating

lessons with little feedback or applied activities,

and having pre-service teachers copy off the

board for extended periods (Tandon & Fukao,

2015).

To tackle this, VVOB developed a 3-year

(2014-2016) programme in close collaboration

with the Cambodian Ministry of Education,

Youth and Sport (MoEYS). One of the

interventions in this programme aimed to

improve both PCK and CK on rational numbers

of mathematics teacher trainers. The impact of

the intervention was measured through a pre-

test post-test design, enriched by qualitative

data collected during lesson observations.

Description of the Intervention

The VVOB-MoEYS Cooperation Program was

designed to strengthen the quality of pre-service

teacher training for primary education in

Cambodia. This intervention fits in with the

overall objective, to strengthen primary school

teacher education in order to improve learning

outcomes in mathematics for all learners. To

ensure the quality of primary teacher education,

PTTCs play an important role in training the

prospective primary teachers. The intervention

programme, therefore, included all mathematics

teacher trainers from 18 PTTCs.

The intervention described in this paper

aimed at improving both PCK and CK on

rational numbers of mathematics teacher

trainers, with a focus on 1) mathematics content

knowledge, 2) the use of representations to

enhance students’ understanding, 3) assessing

pre-service teachers’ learning, and 4) addressing

misconceptions following the concepts of

Shulman (1986). Rational numbers are amongst

the most difficult topics in the elementary school

curriculum, and teaching that topic requires an

adequate knowledge base for teachers to

properly deal with students' difficulties, so it was

selected for the intervention.

A total of 54 mathematics teacher trainers

participated in this intervention. The capacity

building trajectory started in May 2014 and was

completed in August 2016. The course took 23

days consisting of 15-day input training and 8-


day refresher training. Try-out sessions with

pre-service teachers were embedded in all

trainings.

The 23-day course consisted of:

• A 5-day module on rational

numbers, based on the Basic

Education and Teacher Training

manuals (MoEYS, 2011)

• A 5-day module on how to produce

and use teaching aids for math in

primary education and a 4-day

refresher training

• A 5-day module on formative

assessments for primary education

and a 4-day refresher training

The training was facilitated by a core

team of 12 experts in mathematics, attached to

different departments within the Cambodian

Ministry of Education (Teacher Training

Department, Department of Curriculum

Development, Primary Education Department

and Provincial Teacher Training Colleges).

Participants were divided into groups of 25 and

30 participants per two facilitators. PTTC

management in charge of technical teaching

were invited to these training sessions in

addition to math teacher trainers. Beside

sessions on understanding specific math topics,

participants had a chance to tryout the content

with their pre-service teachers and their peers,

to apply peer learning, and to share their

experiences during subject group meetings in

their own Teacher Training College.

The second part of the learning trajectory

consisted of coaching and mentoring sessions,

based on lesson observations. The same

mathematic core team observed the lesson of

teacher trainers in each PTTC. After each lesson

observation, they provided coaching and

mentoring to the teacher trainers to encourages

collaborative and reflective practice. Coaching

allowed teacher trainers to apply their learning

more deeply, frequently, and consistently than

working alone. Each teacher trainer was

observed twice during the learning trajectory.

The focus of these follow-up visits was on:

assessment of learning, addressing the

misconceptions, and using the representation in

the mathematics lesson. In the meantime,

teacher trainers also reflected and translated

content of their lessons into how prospective

teachers apply the instructional strategies. Each

observation was a part of coaching process

consisting of constructive feedback, following

the structures of the 6 feedback steps (MoEYS,

2016). Recordings were also used to analyze the

challenges of math teacher trainers; these issues

were tackled during the following training or

reflection sessions.

Measuring the Impact of the

Intervention

Assessment Tools

1. Pedagogical Content Knowledge and

Content Knowledge Test

Depaepe et al. (2015) developed the test in line

with the Cambodian context to gather

information about the level of mathematics

teacher trainers’ CK and PCK. Depaepe et al.

(2015) defined CK of rational number as

conceptual and procedural knowledge about the

rational numbers domain, as well as, PCK as

knowledge of students' misconceptions and

buggy procedures about rational numbers and of

multiple representations to prevent and/or

remedy these misconceptions and buggy

procedures. The definitions of PCK and CK are

in alignment with Shulman’s conceptualization

of PCK (Shulman, 1986).

The test was composed of 48 questions

with 50% PCK questions and 50% CK

questions. Depaepe et al. (2015) distinguished

between two types of PCK items, namely (1)

knowledge of students’ misconceptions and (2)

knowledge of instructional strategies and

representations. In addition, questions were

categorized in two domains of rational numbers:

fractions (50%) and decimal numbers (50%).

More detailed information is shown in Table 1.

Each item has a maximum score “1”, for an


entirely correct answer. In case of an incorrect

answer, “0” was assigned. On the questions

related to operation, answers were scored “1/2”

if they were partly correct. This is shown in

Table 1.

2. Lesson Observation Checklist

The forms used to observe lessons consisted of

two parts. The first part of the observation

checklist captures parts of the lesson linked to

each of the following categories: (1) Teaching

methodologies, (2) Teaching aids, (3) Learning

content and lesson objective (knowledge, skills,

attitude), (4) Student assessment strategies, (5)

Pupil’s behaviour (level of involvement and

activity), (6) Pupil’s learning outcomes

(remembering/

understanding/applying/analysing/creating/jud

ging) and (7) General lesson characteristics

(structure, build-up, etc). The information

written down in this checklist was used for the

reflection after the lesson.

The core team would use written notes as

the base for the reflection sessions which

followed, including coaching, mentoring and

providing constructive feedback. The coaching

sessions were structured using 6 steps: 1)

introduction, 2) the coaches shares the results of

their teaching, 3) coach give feedback, 4) the

coach ask the coachees to respond to the

feedback, 5) both parties discuss the ways for

improvement, 6) Round up: remaining

questions and making an appointment for the

next meeting (MoEYS, 2016). Each session took

30 minutes and gave the teacher trainer the

chance to reflect on their lesson and teaching

strategy.

The second part of the observation form

consisted of a scoring grid (see snapshot below).

Based on the information collected in part one

and the discussion after the lessons, the core

team gave a score to three selected PCK criteria:

assessment, misconception, and representation.

The assessment part had 4 sub-criteria with a

total score of 12, the misconception part

Table 1

Design of the CK and PCK test: distinguished subdomains and number of items

Domain Sub-domain

CK PCK

Misconception Instructio

n Fractions Concept 4 2 2

Operations Addition 2 1 1

Subtraction 2 1 1

Multiplication 2 1 1

Division 2 1 1

Decimal numbers Concept 4 2 2

Operations Addition 2 1 1

Subtraction 2 1 1

Multiplication 2 1 1

Division 2 1 1

Total 24 24


had 3 sub-criteria with a total score of 9, and

representation had 4 sub-criteria with a total

score of 12. The scoring table described clearly

what needed to be observed for every level, and

for each score. All sub-criteria were scored from

1 to 3, with a score of “1” being the lowest score,

“2” the medium score, and “3” the highest score.

The table also allowed for adding a justification

for the score given, by adding examples in the

‘Proof’ column.

Table 2

Snapshot of scoring grid for PCK criteria ‘Misconception’

Criteria Code Grading scale Proof

Level 1 Level 2 Level 3

Misconceptions

B1 The teacher

doesn’t pay

attention to

mistakes.

The teacher helps

students when

they have made a

mistake by

repeating or

referring to

procedures.

The teacher tries

to understand the

students’ thinking

and helps them

by explaining it in

a different way.

e.g. use of

teaching aids to

support the

weaker students.

B2 The teacher

doesn’t check

prior

knowledge on

the topic.

The teacher

checks the prior

knowledge of

students.

The teacher

checks the

understanding of

prior knowledge.

e.g. Why did you

put both fractions

on the same

denominator?

B3 The only

questions that

are used refer

to knowledge.

The teacher asks

some thinking

questions.

e.g. Why can’t we

just add the

numerators and

denominators?

The teacher asks

many thinking

questions


Pre-Post Test Design

All respondents were assessed using a pre-test

post-test design on PCK/CK. The pretest was

administered in May 2014, the post-test in

August 2016. Both were administered by the

core team of 12 math experts under the

supervision of VVOB project staff. The same

team was also responsible for correcting and

scoring the test. The testing phase was divided

into two parts, the time allowed for each part (24

questions) was 120 minutes. To assure

anonymity, VVOB collected all PCK-test forms

and names were replaced by code before the

correction process started.

In total 54 teacher trainers completed the

pre-test, of those only 33 finished the post-test.

The attrition was caused by different reasons

such as retirement, workplace change, and job

promotion. Besides assessing the tests, the

project team also observed a lesson of each

teacher trainer before, during and after the

intervention to measure the progress and

impact. In total 97 lessons were observed during

thethree-year program. The forms used for these

observations were the same as the observations

tools used during the intervention for the follow-

up visits, but on these occasions, not used with a

coaching purpose. The focus of the pre-post

observations was on using representations,

misconceptions, and assessment.

Data Analysis

Descriptive and inferential statistics were

conducted on data set. Percentage and frequency

were used to describe respondents’ information

background related to the qualifications, years of

experience, socio-demographic information, and

the progress of achievement scores from lesson

observation focusing on how to apply the

formative assessment techniques, to addressing

the misconceptions, and use of the

representation. Moreover, another achievement

was measured by pre and post-test of PCK. A

paired t-test was performed to compare the

mean score of both tests. Achievement was

measured to determine if post-test scores

increased significantly compared to pre-test

scores, at significant level = 0.05 .

Results

As described above the impact of the

intervention was measured through a pre-test -

post-test design; a group of 33 teacher trainers

completed both tests. The paired-sample T-test

found that the mean of the overall score on the

post-test (M=33.2, SD=7.5) of teacher trainers is

significantly higher than their score in pre-test

(M=27.3, SD=7.5), with significant increase of

5.9 (95%CI: 3.94-7.93, p<0.001, t(32)=6.044).

The preliminary analyses show a great

disparity between the scores of the teacher

trainers. Descriptive data analysis showed that

female teacher trainers performed better than

their male peers in both pre-test and post-test,

however this difference was not significant.

We also saw that the mean scores of

(young) teacher trainers with less years of

teaching experience, was higher than their

senior peers in both pre-test and post-test. A

clarification for this result could be found in the

educational background of the teacher trainers.

All young teacher trainers had graduated from

university with a bachelor’s degree, while most

of the senior teacher trainers graduated from a

2-year programme at a teacher training college.


Table 3

Mean score by gender, years of experience and qualification

Categories of participants Number

Mean score

before

Mean score

after

intervention

(SD)

intervention

(SD)

Gender

Male 28 26.8 (8.0) 32.1 (7.9)

Female 5 28.2 (5.7) 36.6 (5.1)

Years of experience

Less or equal to 5 years 6 27.6 (6.5) 35.2 (5.0)

Equal or more than 6 years 27 27.2 (7.8) 32.8 (7.9)

Qualification

Master 10 27.3 (8.6) 35.9 (8.0)

Bachelor 13 28.5 (7.3) 33.7 (5.7)

Teacher Training certificate 10 25.7 (7.0) 29.9 (8.5)

Overall score* 33 27.3 (7.5) 33.2 (7.5)

*Mean score after intervention is significantly higher than before intervention (p<0.001)

When we looked closer at the differences

between PCK and CK tests, we noticed teacher

trainers scored better in both on the post-test

compared to the pre-test. The paired sample t-

test showed a significant increase on both mean

score of CK (p<0.001, t(32)=4.165) items with

95% confident interval of difference: 1.03-3.02

and PCK (p<0.001, t(32)=5.493) items with 95%

confident interval of difference:2.45-5.35 after

intervention. We noticed the scores on

pedagogical content knowledge items increased

much more (t(32)=2.6, p=0.014) , compared to

the scores on related to pure content knowledge

items. During coaching sessions, teacher trainers

indicated that they had more difficulty

answering the questions related to PCK than the

CK items. Looking closer at the responses within

the PCK items, we see teacher trainers struggled

more with instructional strategies and

representation (mean=6.3, SD=2.7) than

explaining students’ misconception (mean=8.0,

SD=2.4) after intervention(t(32)=-4.64,

p<0.001). More details can be found in Table 4.


Table 4

Comparison of PCK test items and CK test items (pre-test and post-test)

Test items categories

Mean score

(SD)

Mean

Difference

(SD)

95% CI P

Pre-test Post-test

PCK items 10.4 (4.5) 14.3 (4.7) 3.9 (4.1) 2.5-5.4 <0.001*

Knowledge of

students’

misconception

5.6 (2.8) 8.0 (2.4) 2.4 (2.7) 1.5-3.4 <0.001*

Knowledge of

instructional

strategies and

representation

4.9 (2.3) 6.3 (2.7) 1.5 (2.3) 0.7-2.3 0.001*

CK items 16.8 (3.8) 18.9 (3.2) 2.0 (2.8) 1.0-3.0 <0.001*

Concept 3.9 (2.1) 5.4(1.8) 1.5 (0.4) 0.8-2.2 <0.001*

Operation 13.0(2.3) 13.1(2.0) 0.1 (1.9) -0.6-0.8 0.7

* Statistically significant increase, at significant level = 0.05

Table 5

Progress on PCK/CK of fractions

Fraction test items

Mean score

(SD)

Mean

Difference

(SD)

95% CI P

Pre-test Post-test

PCK items 4.2 (2.2) 6.8 (2.6) 2.6 (2.1) 1.8-3.3 <0.001*

Knowledge of

students’

misconception

2.6 (1.3) 3.8 (1.6) 1.4 (1.5) 0.6-1.7 <0.001*

Knowledge of

instructional

strategies and

representation

1.6 (1.1) 3.0 (1.5) 1.4 (1.3) 1.0-1.9 <0.001*

CK items 7.8 (2.1) 9.0 (2.1) 1.2 (1.9) 0.6-1.9 0.01*

Concept 1.7 (1.3) 2.6 (1.2) 1.0 (1.6) 0.4-1.5 0.001*

Operation 6.2 (1.4) 6.1 (1.5) –0.2(1.4) –0.7-0.4 0.5

* Statistically significant increase, at significant level = 0.05


Table 5 provides a closer look at the

differences between progress made related to

understanding and teaching fractions compared

to teaching and understanding decimals

numbers. Table 5 shows the scores on both PCK

and CK items on the questions about fractions.

Teacher trainers performed significantly better

on both PCK (t(32)=3.73, p=0.01)and CK

items(t(32)=6.9, p<0.001) for fractions after

intervention. However, no significant increase

was found if we consider the scores for CK items

related to operations with fractions. This can be

explained by already high scores at the start of

the intervention, approximately 6 over the scale

of 8, in both pre-test and post-test.

Teacher trainers had more difficulty with

fraction PCK items than fraction CK items in

both pre-test (t(32)=-7.516, p<0.001) and post-

test (t(32)=10.34, p<0.001). However, mean

scores of fraction PCK (t(32)=6.901, p<0.01)

items and fraction CK (t(32)=3.73 ,p=0.01 items

were significantly higher after intervention.

Table 6 shows that teacher trainers

performed significantly better on decimal CK

items (t(32)=2.548, p=0.016) and decimal PCK

items (t(32)=2.644 , p=0.013) after receiving

capacity development. Nevertheless, they still

struggled more with PCK items related to

decimal numbers than CK items in both pre-test

(t(32)=-6.888, p<0.001)) and post-test (t(32)=-

7.104, p<0.001)). Teacher trainers performed

better on knowledge of students’ misconception

(t(32)= 3.954 , p<0.001) after intervention, but

they made no significant progress regarding the

knowledge of instructional strategies and

representations (t(32)=0.99 , p=0.922). The

high score of CK on operation with decimal

numbers (almost 7 on a maximum score of 8) is

remarkable, although it is not statistically

significant.

Table 7 presents the most challenging PCK

items and CK items for teacher trainers, even

they have taken a training course on rational

number. The teacher trainers had more difficulty

putting the fraction into words, and matching

this with the corresponding section in the word

problem. This challenge indicated that they had

limited knowledge about how to translate real-

life word problems into number sentences or

vice versa, for example PCK item 1 and CK item

Table 6

Progress on PCK/CK of decimal numbers

Decimal test items

Mean score

(SD)

Mean

Difference

(SD)

95% CI P

Pre-test Post-test

PCK items 6.2 (2.9) 7.6 (2.5) 1.3 (2.9) 0.3-2.3 0.013*

Knowledge of

students’

misconception

2.9 (1.8) 4.2 (1.4) 1.3 (0.3) 0.6-2.0 <0.001*

Knowledge of

instructional

strategies and

representation

3.3 (1.6) 3.3 (1.6) 0.0 (1.8) –0.6-0.6 0.922

CK items 9.0 (2.1) 9.8 (1.6) 0.8 (1.8) 0.2-1.5 0.016*

Concept 2.2 (1.1) 2.8 (1.1) 0.5 (0.9) 0.2-0.9 0.001*

Operation 6.8 (1.5) 7.1 (1.0) 0.3 (1.4) –0.2-0.8 0.271

* Statistically significant increase, at significant level


Table 7 The most difficult questions of PCK items and CK items after intervention

PCK/CK Items % correct

response

PCK

Given: − =3 1 7

5 4 20

1.

Indicate and explain for each of the below mentioned word problems whether

you would use them in your classroom to contextualize the above-mentioned

operation.

a) 3

5 of a cake was used by dad. Sopheak and Sophy eat together

1

4 of the

remaining part of the cake. How much of the cake have they eaten?

b) To fill a water basin we need1

4 of a completely filled open well. Today

the open well is only filled for 3

5. How much water remains in the

open well after the water basin is filled?

c) When frying vegetable dad uses3

5 of a small bottle of chili sauce and

1

4

of a small bottle of soya sauce. How much chili and soya sauce

remains?

15.2

2. These are illustrations of elementary students’ solutions.

Samnang’s solution Champey ‘s solution Malis’s solution

Determine the right or wrong solution. In case of a wrong solution, write down

the presumable student’s reasoning.

24.4

CK

1. If the rectangle below is6

5 of the surface of the original shape, draw the

original shape.

45.5

2. Write down and solve the mathematical operation with fractions that fits the

following problem:

Somaly made 4

5 liter of fresh fruit juice. She gave

1

4 to her mother.

How many liter of fresh fruit juice did her mother receive?

45.5


.

2 in table 7. After intervention, roughly 15% of

teacher trainers answered PCK item 1 correctly

and 45% of teacher trainers correctly answered

CK item 2. Understanding how to address

students’ misconception or difficulties remained

a challenge for mathematics teacher trainers

after the intervention. As a result, approximately

one-fourth of them could explain students’

reasoning or misconception, when students

provided a wrong answer.

After intervention, roughly 15% of teacher

trainers answered PCK item 1 correctly and 45%

of teacher trainers correctly answered CK item 2.

Understanding how to address students’

misconception or difficulties remained a

challenge for mathematics teacher trainers after

the intervention. As a result, approximately one-

fourth of them could explain students’ reasoning

or misconception, when students provided a

wrong answer.

In addition, teacher trainers had

difficulties understanding the concept of

'fraction'. Only 45.5% could draw the original

shape of 6

5 in the rectangle correctly, CK

question 1.

To follow up on the progress of teaching

mathematics and coach the teacher during the

implementation of the newly acquired skills, 94

mathematics lessons were observed by the

expert teams. Each teacher trainer was observed

4 times (2 times as part of the pre-post test, and

2 times as part of the individual coaching

sessions) by a team of two experts.

Descriptive analysis revealed that scores of

the lesson observations gradually increased from

roughly 69% of total score of 33 at the start of

the intervention to 92.4% at the end. Teacher

trainers improved most in the field of assessing

their students. They also made progress in using

representations and detecting students’

misconceptions, but the upward growth trend

was less pronounced. The percentage of

achievement score in each criterion increased in

second lesson observation in comparison with

the first. Then the achievement score decreased

eventually in relation to second observation. The

score of the final lesson observation gradually

increased in comparison to the previous three.

Fluctuations in the score of the third

observations were caused by an increase of

teacher trainers in the cohort. Those additional

teacher trainers were not mentored and coached

by the expert team in the first and second lesson

observations so their achievement score from

lesson observation were lower than their peers

included from the beginning in the learning

trajectory. This indicated once more the

importance of coaching and mentoring for

strengthening teachers’ capacity. Teachers also

confirmed during the evaluation of the

programme how beneficial the coaching sessions

after each lesson observation were for improving

their future teaching.

Looking at the data in Table 3 and Table 6,

it becomes clear that the use of representations

during math lessons is the most challenging area

of PCK. Teacher trainers underperformed during

lesson observations, and on the test items

related to using representations. It is

encouraging that teacher trainers’ capacity in

this area increased compared to their

performance at the beginning of programme.

Discussion and Conclusions

Our findings highlight the urgent need to

improve the preparation of future teachers with

respect to subject-matter knowledge (CK and

PCK). We described an intervention to improve

teacher trainers’ content knowledge (CK) and

pedagogical content knowledge (PCK) on

fractions and rational numbers. The results

revealed gaps in teacher trainers' CK and PCK

for fractions and decimal numbers. Most of

these gaps were significantly reduced by the end

of the intervention. After the intervention, 91%

of the total teacher trainers who were observed

by the math expert team had significantly

increased their score on the PCK test and 94%

had improved their teaching strategy in at least

two of three criteria (representation,

misconception, and assessment). The results


Table 8 Progress on mathematics teaching of teacher trainers

Date Follow

up

Number

Number of lesson

observations

Score achievement of PCK areas total

achievement

(% of 33) Represents

(% of 12)

Misconceptions

(% of 9)

Assessments

(% of 12)

1. Apr-2014 11 (75.0) (58.9) (68.9) (68.9)

2. Apr-2015 11 (90.2) (83.8) (85.6) (86.8)

3. Jan-2016 36 (83.5) (81.9) (88.8) (85.0)

5..Jul-2016 36 (89.2) (91.0) (96.8) (92.4)

confirmed the importance of coaching and

mentoring as key elements of success in

strengthening the capacity of teacher trainers.

Limitations

Pre- and post-tests (as well as the intervention

itself) were limited to fractions and decimal

numbers. There was no control group, which

limits the generalizability of the findings. Studies

assessing teachers’ competence in other domains

are required as well as within the domain of

mathematics. The high turnover of teacher

trainers during the intervention, made

comparison of pre- and post-test results

difficult, as the size of the sample became too

small to make certain conclusions. Another

limitation was the creation of the assessment

tool. Since no valid PCK test was available for

Cambodian teachers, we used a validated PCK

test developed by the University of Leuven

(Belgium). Giving priority to the reliability of the

test, there was little room for modifications of

the items, resulting in less opportunity to adjust

the items to the Cambodian context. Translation

challenges (Dutch-English-Khmer) also

complicated the understanding of the items for

test administrators and the participants. Finally,

by tailoring the learning trajectory to the needs

of the teacher trainers, not all math topics

tackled during the training were part of the

standard PCK test. Conversely, some math items

included in the assessment tool, were not part of

the learning trajectory.

It would be interesting for future

interventions to study the relationship of PCK

and CK of teacher trainers with the learning

outcomes and teaching skills of pre-service

teachers. Research has shown that coaching

allows teachers to apply their learning more

deeply, frequently, and consistently than

teachers working alone, and we strongly believe

coaching is important to make teacher trainers

reflect and adjust their teaching practices.

However more research is needed on how

coaching supports teacher trainers to improve

their capacity to reflect and apply their learning

to their work with pre-service teachers and in

their work with each other.

Notes

1. VVOB stands for Vlaamse Vereniging voor

Ontwikkelings-samenwerking en technische

Bijstand Dutch It means Flemish

Association for Development Cooperation

and Technical Assistance.

2. The Association of Southeast Asian Nations

is a regional intergovernmental organization

with the purpose of facilitating economic

growth, social progress

and cultural development that includes ten

Southeast Asian countries.


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About Author (s)

Leap Van, MSc, is an education programme

coordinator for mathematics education of VVOB

Cambodia. His role is to provide technical

support to build capacity of teacher trainers on

teaching primary mathematics including STEM

education in Teacher Education College. Also he

is a lecturer of biostatistics of faculty of science

and technology at International University,

Cambodia. He used to be a math-physic teacher

at high school for 10 years.

Ms. Sokalyan Mao is an education project

manager of Finn Church Aid Foundation, a

Finnish non-Government organization. She is

currently managing two projects career guidance

and counselling and the Dream School Project.

She had worked for VVOB- Education for

Development for four years as a programme

coordinator. She was in charge of two

programmes focusing on mathematics and

science including STEM. Before she worked for

development organizations, she was a teacher at

a secondary school for 10 years. Sokalyan earned

her master’s degree in Education from the

Victoria University of Wellington, New Zealand,

and a master’s degree in Rural Development

Management from Khon Kaen University,

Thailand.

Ms. Veerle Cnudde is currently working as

Policy Advisor at the Department of Foreign

Affairs for the Government of Flanders. She

worked for 20 years at VVOB- Education for

Development managing and implementing

education programs in developing countries,

including Cambodia, Zambia and Chile. Veerle

received her master’s degree in Educational

Science from the University of Ghent. Her

interests and expertise lie in strengthening

capacity and ownership to improve the quality of

education.

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